Statistical localization: From strong fragmentation to strong edge modes

Certain disorder-free Hamiltonians can be nonergodic due to a strong fragmentation of the Hilbert space into disconnected sectors. Here, we characterize such systems by introducing the notion of “statistically localized integrals of motion” (SLIOM), whose eigenvalues label the connected components of the Hilbert space. SLIOMs are not spatially localized in the operator sense, but appear localized to subextensive regions when their expectation value is taken in typical states with a finite density of particles. We illustrate this general concept on several Hamiltonians, both with and without dipole conservation. Furthermore, we demonstrate that there exist perturbations which destroy these integrals of motion in the bulk of the system while keeping them on the boundary. This results in statistically localized strong zero modes, leading to infinitely long-lived edge magnetizations along with a thermalizing bulk, constituting the first example of such strong edge modes in a nonintegrable model. We also show that in a particular example, these edge modes lead to the appearance of topological string order in a certain subset of highly excited eigenstates. Some of our suggested models can be realized in Rydberg quantum simulators.

Figure 1

Statistical locality of SLIOMs. Expectation value $\langle \psi |{{\mathcal{O}}_i^k}^{†}{\mathcal{O}}_i^k|\psi \rangle$ for the string operators appearing in the definition of SLIOMs ${\widehat{q}}_k={\sum }_i{\mathcal{O}}_i^k$, see Eq. (4). The averages are performed over: a Haar random state $\left|\psi \right.〉$ in the full Hilbert space with average filling fraction $\nu =2/3$ in (a) and (b), and a random state with a fixed filling fraction $\nu =N_F/L=1/2$ in (c) and (d), evaluated analytically via Eqs. (5) and (7), respectively. [(a) and (c)] In both cases, the $k\mathrm{th}$ particle, is statistically localized around the average position $\overline{i}=k/\nu$. (Inset) The statistical localization of the boundary SLIOMs ${\widehat{q}}_{\ell },{\widehat{q}}_r$ [defined in Eqs. (9) and (10)], with at least exponential decay towards the bulk. [(b) and (d)] When considering SLIOMs in the bulk, $k\propto L$ [$k=L/2$ for (c) and $k=N_F/2=L/4$ for (d)], the width of the distribution scales as $\sqrt{L}$, and the height as $1/\sqrt{L}$.

Figure 2

Autocorrelations for the $t\text{−}{J}_z$ model in the bulk (a) Mazur bound (8) on autocorrelations in the bulk, at $j=L/2$, decays as $\propto L^{-1/2}$ as a function of the system size $L$. (b) The same bound, shown for a fixed $L=600$, decays as $\propto j^{-1/2}$ as a function of the distance $j$ from the boundary. (c) The long-time average of spatially resolved correlations, computed numerically for small chains (and averaged between times $t=50$ and 100), shows a persistent peak, instead of the complete spreading expected from thermalization.

Figure 3

Diagonal matrix elements in the $t\text{−}{J}_z$ model. Expectation value of the average nearest-neighbor antiferromagnetic correlations in eigenstates of $H_{t\text{−}{J}_z}$ with global quantum numbers $N_F=L/2$ and ${\sum }_j{S}_j^z=0$, and open boundary conditions. For the system sizes shown ($L=8,12,16$), the distribution becomes wider with increasing system size, while asymptotically it is expected to narrow as $\sim L^{-1/4}$. This is a consequence of the strong fragmentation labeled by the SLIOMs, and is in contrast with ETH, which predicts an exponentially narrow distribution.

Figure 4

Bulk vs edge autocorrelations. Connected infinite temperature autocorrelation function for the center site $i=L/2$ and at the left boundary $i=1$ for system sizes $L=11,13,15$. (a) In the $t\text{−}{J}_z$ model (1), which conserves both bulk and boundary SLIOMs ${\widehat{q}}_k$, the edge autocorrelator shows infinite coherence times while in the bulk it decays to a value $\propto L^{-1/2}$, which is anomalously large but vanishing in the thermodynamic limit. (b) Once the perturbation (12) is added, SLIOMs in the bulk are broken and the bulk autocorrelations decay to the value $\propto 1/L$ expected for thermalizing systems. The boundary SLIOMs ${\widehat{q}}_{\ell },{\widehat{q}}_r$, on the other hand, are still conserved, leading to a finite long-time value for autocorrelations at the edge, well approximated by the analytical lower bound (dashed horizontal line).

Figure 5

Mapping from spin-1 chain to bond spins and defects. (a) A charge configuration with alternating signs can be mapped to spin-1/2 variables on the bonds. (b) For a generic configuration, one also has to introduce defects, living on sites, whenever a charge would violate the rule of alternating signs. Note that defects with neighboring bond spins pointing to the right (left) correspond to positive (negative) charges in the original.

Figure 6

Hopping of defects. To maintain the constraints, when a defect hops it has to flip a bond spin to its right, making its dynamics asymmetrical. In the original variables, this process is equivalent to emitting/absorbing a dipole from the right.

Figure 7

Labeling of connected sectors in the original variables. Charges that have the same sign as the ones to their left (circled) correspond to defects, whose total number ($N^d$) and pattern is conserved by the Hamiltonian $H_3$. Moreover, the dipole moment ${\widehat{P}}_k$ within each region between two subsequent defects (including the defect on the left but not the one on the right, as indicated by the brackets) is also independently conserved, such that the total dipole becomes $P={\sum }_{k=0}^{N^d}{\widehat{P}}_k$.

Figure 8

Entanglement growth for $H_3$. The saturation value of the half-chain entanglement at long times for the dipole-conserving Hamiltonian $H_3$ [Eq. (14)] for initial product states in the $S^z$ basis can be understood from the emergent conservation of the number of defects $N^d$, along with the SLIOMs ${\widehat{P}}_k$ introduced in Sec. 3b2. The former implies a block-diagonal structure of the reduced density matrix ${\rho }_A$ on region $A$ (chosen to be half the chain), of the form ${\rho }_A={\oplus }_{N_A^d=0}^{N^d}{\rho }_A\left(N_A^d\right)$. However, due to the kinetic constraints on the mobility of defects (see main text), only a few of these blocks are nonvanishing, those where $N_A^d$ is close to its value in the initial state. The additional conservation of dipole moment within a region between defects $\left\{{\widehat{P}}_k\right\}$ further block diagonalizes ${\rho }_A\left(N_A^d\right)$, most of which are again zero.

Figure 9

Spatial distribution of SLIOMs for energy eigenstates. Spatial distribution of the expectation value $\langle \psi |{{\mathcal{O}}_i^k}^{†}{\mathcal{O}}_i^k|\psi \rangle$, for the ground state (left) and an excited state (right) within the sector $N_F=L/2$, $S_{\text{tot}}^z=0$. The excited state is randomly picked from within a sector with a fixed spin pattern (see main text). Both states correspond to (partially) localized distributions. For the excited state, this is close to the Haar average (dashed lines), while for the ground state the distribution is more tightly localized.

Figure 10

Time averaged correlations vs. their conjectured values. The dots (connected by narrow dashed lines) show the long-time average (averaged between times $t=50$ and 100) of the correlator ${\langle S_j^z\left(t\right)S_i^z\rangle }_{\beta =0}$, while the solid lines represent $C_{ij}\left(\infty \right)$, defined by the formula (D1). This is a lower bound near the origin, but becomes smaller then the numerical value in the tails (i.e., the observed distribution is actually narrower than the prediction). However, the two curves approach each other as system size is increased. This is shown by the inset, where the blue dotted curve represents ${\sum }_i{\left[\frac{1}{T}{\int }_{50}^{100}dt{\langle S_j^z\left(t\right)S_i^z\rangle }_{\beta =0}-C_{ij}\left(\infty \right)\right]}^2$ as a function of $L$, approximately decreasing as $1/L$ (red dashed line).

Figure 11

Thermalization in the $t\text{−}{J}_z$ model with closed boundaries Left: expectation values of nearest-neighbor antiferromagnetic correlations in eigenstates for $N_F=L/2$, $S_{\text{tot}}^z=0$ for different system sizes. The distribution has a width that does not decrease with system size. Right: time average (between times 50 and 100) of the spatially resolved spin-spin correlations at infinite temperature. While there is a small peak around the origin remaining for the available system sizes, the correlations mostly spread out over the whole chain and take values $\propto 1/L$, unlike the case of an open chain shown in Fig. 2.

Figure 12

Scaling of the saturation value of the entanglement entropy with system size. We show the ratio of the saturation value of the entanglement entropy $S\left(\infty \right)$ and the Page value $S_{\text{Page}}=ln\left(3\right)L/2-1/2$, for for $H_3$ (red circles), $H_3+H_4$ (blue stars), and $H_{t\text{−}{J}_z}$ (green squares).